Mapping a Sphere to a Plane
Date: 11/28/2001 at 12:15:18 From: Candice Subject: Maps of the world Maps of the world are always distorted in some way when put on a flat map instead of globe. Why? I have to answer this question on a college level and an elementary level. I would explain this to elementary students with an orange peel. We can see that an orange peel covers the whole orange before we peel it. Then after we peel it we can see that the peel would not lie flat on a table. I 'm not sure where to go from here. I'm not sure which facts of geometry I should use to investigate this on a college level. I think that using the formula of the surface area of a sphere might work, but I'm not sure how to use this formula to answer the question. Thanks, Candice
Date: 11/28/2001 at 13:18:31 From: Doctor Tom Subject: Re: Maps of the world Hi Candice, The proof that it's impossible uses differential geometry, but I can perhaps indicate the idea behind the proof. Any smooth surface has a "curvature" at every point. In the case of the plane, the curvature happens to be zero everywhere. The curvature may not be exactly what you think it is. On a cone, away from the tip, the curvature is also zero. (There are different sorts of curvature, but the type I'm interested in has zero curvature.) And, in fact, a cone could be slit and flattened out on a plane. You can locally measure this curvature if you "live" on such a surface as follows: Take a point and put a nail in the ground. Take a piece of string of length 1 tied to the nail, and draw a "circle" by connecting the other end to a pencil and dragging it around as far from the nail as possible. Now measure the length of the circle. If it's 2*pi, the curvature is zero. If it's bigger, the curvature is negative. If it's smaller, the curvature is positive. A sphere obviously has positive curvature everywhere since the circle on the surface of the sphere is smaller than it would be on a plane. (If you have trouble imagining this, think of a 1-foot string on a 2-foot sphere.) Things like saddles are the opposite - the circle on a saddle would be bigger than on a plane. If you try to imagine mapping a sphere to a plane in a way that preserves all lengths, you're obviously out of luck. The the points on the plane will all have to be the same distance away from the image of the center, but then you will have changed the lengths around the circle, so the scaling is wrong there. You can map a small enough piece of a sphere to the plane with an arbitrarily small error, but as you decrease the allowable error, the map gets smaller and smaller, and none of them are mathematically perfect. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/
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